During the cutting-off and turning of workpieces, friction and forming lead to a strong evolution of heat, as a result of which the workpiece, but also the cutting tool used, is greatly heated. However, the high temperatures of the cutting tool lead to increased wear, resulting in costs. For this reason, the cutting insert is cooled with the aid of a coolant during the machining of the workpiece. To date, it has been conventional for a stream of coolant to be guided to the cutting insert, in particular to the main lip of the cutting insert, through nozzles or ducts in or on the tool holder in which the cutting insert is mounted. This has the problem that the stream of coolant is directed to a strong extent, and therefore merely the portion of the cutting insert which is directly impacted by the stream of coolant is cooled.